Motivate the Math

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Motivate the Math

What is the math that enables cryptography?

قبل ٤ أيام

MTM08: Math Culture and Logarithms

Square and Multipply Algorithm https://www.youtube.com/watch?v=cbGB__V8MNk Fundamentals npub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99g AverageGary npub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9

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٢٦ فبراير

MTM07: Proofs / How Do You Know You What You Think You Know?

Proofs https://www.youtube.com/watch?v=HIkIqt_ytdc Fermat's Little Theorem https://mathworld.wolfram.com/FermatsLittleTheorem.html Euler's Function https://en.wikipedia.org/wiki/Euler's_totient_function Elliptic Curves: Point Addition https://www.rareskills.io/post/elliptic-curves-finite-fields Fundamentals npub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99g AverageGary npub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9

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MTM06: Proof By Mathematical Induction: A Validation Cheatcode

Proof by Mathematical Induction https://youtu.be/Tm2PJPvAULs?si=H_RJ5rmVeyPDYM9W https://youtu.be/KW5k7ZsQmwo?si=8rEdf2dUcTw74QZ5 Understanding Cryptography https://www.youtube.com/watch?v=2aHkqB2-46k Fundamentals npub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99g AverageGary npub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9 In this episode, we dive deep into the fascinating world of elliptic curves and their significance in cryptography. We start by discussing the basics of elliptic curves, particularly focusing on the polynomial equation y² = x³ + 7, which is crucial for Bitcoiners. We explore how operations on these curves, like adding points, form a group and why this concept is important. We then delve into the textbook by Neil Koblitz, which highlights the importance of elliptic curves in cryptography. The discussion transitions into the axioms of groups, such as closure, associativity, identity, and inverses, and how these relate to elliptic curves. Our conversation takes a turn towards Fermat's Little Theorem and its application in cryptography, particularly in computing inverses in finite fields. We explore how this theorem simplifies calculations with large numbers and its implications for public key cryptography. We also touch on the Diffie-Hellman key exchange, explaining how it enables secure communication over the internet by deriving a shared secret without exposing private keys. Throughout the episode, we emphasize the importance of understanding these mathematical concepts to grasp the underpinnings of cryptographic systems, especially in the context of Bitcoin and other cryptocurrencies.

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١٩ فبراير

MTM05: Elliptic Curves and.Fermat's Little Theorem

2^173(mod5) = (2^4)^43 * 2^1 = 1^43 * 2^1 = 1 * 2 = 2 2^4(mod5) = 1 because 16(mod5) = 1 Fermat's Little Theorem https://mathworld.wolfram.com/FermatsLittleTheorem.html Euler's Function https://en.wikipedia.org/wiki/Euler's_totient_function Elliptic Curves: Point Addition https://www.rareskills.io/post/elliptic-curves-finite-fields Diffie-Hellman Illustration https://www.youtube.com/watch?v=NmM9HA2MQGI Fundamentals npub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99g AverageGary npub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9 In this episode, we dive deep into the fascinating world of elliptic curves and their significance in cryptography. We start by discussing the basics of elliptic curves, particularly focusing on the polynomial equation y² = x³ + 7, which is crucial for Bitcoiners. We explore how operations on these curves, like adding points, form a group and why this concept is important. We then delve into the textbook by Neil Koblitz, which highlights the importance of elliptic curves in cryptography. The discussion transitions into the axioms of groups, such as closure, associativity, identity, and inverses, and how these relate to elliptic curves. Our conversation takes a turn towards Fermat's Little Theorem and its application in cryptography, particularly in computing inverses in finite fields. We explore how this theorem simplifies calculations with large numbers and its implications for public key cryptography. We also touch on the Diffie-Hellman key exchange, explaining how it enables secure communication over the internet by deriving a shared secret without exposing private keys. Throughout the episode, we emphasize the importance of understanding these mathematical concepts to grasp the underpinnings of cryptographic systems, especially in the context of Bitcoin and other cryptocurrencies.

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٦ فبراير

MTM04: Levelling Up on Bitcoin's Supply Cap

Bitcoin's Issuance Schedule https://blog.lopp.net/how-is-the-21-million-bitcoin-cap-defined-and-enforced/ What is a Geometric Series https://en.wikipedia.org/wiki/Geometric_series Calculating the Sum of a Geometric Series https://www.youtube.com/watch?v=PqXAjCXYbNk Fundamentals npub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99g AverageGary npub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9 In this episode, we delve into the complexities of mathematics and its profound impact on our understanding of the world, particularly in the realm of cryptography. We discuss the challenges of making complex mathematical concepts accessible and why it's crucial not to blindly trust mathematical protocols without understanding their foundations. Our conversation explores the role of mathematics as a language that explains the workings of the world, emphasizing its importance beyond mere numbers. We also touch on the human brain's ability to recognize patterns and how this relates to our perception of reality and survival instincts. The discussion extends to neuroplasticity and the potential to "rewire" our brains through practice and repetition, drawing parallels between physical and mental fitness. Our exploration includes a deep dive into the concept of geometric series, particularly in the context of Bitcoin's block subsidy and its mathematical underpinnings. We explain how numbers can be represented as polynomials and the significance of fields and rings in mathematics, highlighting the unique properties of binary systems. Throughout the episode, we emphasize the importance of teaching and sharing knowledge to deepen understanding, and we encourage listeners to engage with mathematical concepts actively. We also reflect on the challenges and rewards of discussing complex topics and the personal growth that comes from pushing intellectual boundaries.

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٣٠ يناير

MTM03: Modular Arithmetic: The Queen Bee

What is a Relation https://www.youtube.com/watch?v=1v0qH4l9A2c&list=PLg8ZEeSiXsjgoQJzRcq60GjK0UrkMsA3-&index=12 What is an Equivalence Relation https://www.youtube.com/watch?v=o-PhSZztHC0&list=PLg8ZEeSiXsjgoQJzRcq60GjK0UrkMsA3-&index=13 Modular Arithmetic (a little advanced but its good support until I find something simpler) https://www.youtube.com/watch?v=d-n92Ml1iu0&list=PLDDGPdw7e6Ag1EIznZ-m-qXu4XX3A0cIz&index=78 Fundamentals npub12eml5kmtrjmdt0h8shgg32gye5yqsf2jha6a70jrqt82q9d960sspky99g AverageGary npub160t5zfxalddaccdc7xx30sentwa5lrr3rq4rtm38x99ynf8t0vwsvzyjc9 In this episode, we delve into the intricate world of cryptography, focusing on the mathematical foundations that underpin modern cryptographic systems. We start by exploring the AES chapter from the "Understanding Cryptography" PDF, discussing the layers beyond the mathematical sections. Our conversation highlights the importance of understanding both the cryptographic implementations and the mathematical relevance behind them, particularly in the context of public and private key cryptography, such as RSA and elliptic curve cryptography. We emphasize the significance of modular arithmetic, describing it as a fundamental substrate for cryptography. The discussion includes personal anecdotes about internalizing modular arithmetic and the continuous learning journey in understanding its applications. We also touch upon the Euclidean algorithm and its role in finding the greatest common divisor, which is crucial for cryptographic functions. The episode further explores the concept of cyclic groups and their relevance in cryptography, particularly in how they can reorder elements to enhance security. We discuss the importance of understanding linear combinations and equivalence relationships, which are foundational in mathematical modeling and cryptographic analysis. We also address some errata from previous episodes, clarifying definitions related to binary operations and cyclic groups. The conversation is enriched with practical examples, such as prime factorization and its role in cryptographic algorithms. Finally, we express gratitude to our listeners and those who have supported the podcast through boosts, encouraging them to engage with the material at their own pace and to explore additional resources provided in the show notes.

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٢٥ يناير · إضافة

MTM02.1 AI Recap: Groups & Structures

Motivate the Math

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٢٣ يناير

MTM02: Exploring Groups and Mathematical Structures

Wrath of Math: https://www.youtube.com/watch?v=VzsAehzmjrU&list=PLztBpqftvzxVvdVmBMSM4PVeOsE5w1NnN A Book of Abstract Algebra: https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178/ In this second episode of "Motivate the Math," we dive into the foundational concepts of group theory, a crucial element in understanding cryptography and mathematics. We start by addressing feedback from our first episode, emphasizing the importance of making math accessible and correcting any inaccuracies. We introduce the concept of an errata to keep ourselves accountable and transparent. We then explore the definition of a group in mathematics, discussing its four key axioms: closure, associativity, identity, and inverse. We explain how these properties are essential for a set of elements to be considered a group and why this matters in the broader context of math and cryptography. We also touch on the concept of commutativity, or Abelian groups, and introduce the idea of cyclic groups. Throughout the episode, we reflect on the feedback and support from our listeners, sharing some of the boosts and messages we've received. We discuss the motivation behind the podcast and the desire to make complex mathematical concepts more digestible and engaging. As we wrap up, we hint at future topics, including rings and fields, and the importance of understanding these concepts for cryptography. We also discuss the need for additional resources and problem-solving sessions to support the podcast's content, emphasizing the collaborative nature of learning and the journey we're on together.

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١٤ يناير

MTM01: Motivating the Math

In this episode, Average Gary and Fundamentals dive into the fascinating world of mathematics, exploring its critical role in understanding Bitcoin and cryptography. They begin by discussing the motivation behind learning math, especially in the context of Bitcoin, and why it is essential for achieving personal sovereignty and a deeper understanding of cryptographic principles. The hosts explore the common phrase "do the math" often heard in Bitcoin circles, unraveling the complexities behind it and emphasizing the importance of understanding the mathematical foundations of Bitcoin's protocol. They delve into the concept of finite fields, elliptic curves, and the significance of cryptographic algorithms, breaking down these complex topics into more digestible concepts. Average Gary shares his journey into the world of math, driven by a desire to understand Bitcoin at a deeper level, and how this led him to explore cryptography and number theory. Fundamentals adds insights into the historical context of cryptography, its evolution, and its current applications in the Bitcoin space. The episode also touches on the importance of self-learning and the availability of resources today that make it easier to study math and cryptography independently. They discuss the potential vulnerabilities in the Bitcoin ecosystem due to a lack of widespread understanding of cryptography and the need for more education in this area. Listeners are invited to join this journey of exploration, as the hosts aim to demystify math and cryptography, empowering individuals to gain a higher level of understanding and confidence in their reasoning and decision-making related to Bitcoin and beyond.

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What is the math that enables cryptography?

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Motivate the Math